Grade 12 Pre-Calculus Mathematics Achievement Test. Marking Guide

Transcription

1 Grade 12 Pre-Calculus Mathematics Achievement Test Marking Guide June 2017

2 Manitoba Education and Training Cataloguing in Publication Data Grade 12 pre-calculus mathematics achievement test. Marking guide. June 2017 This resource is available in print and electronic formats. ISBN: (print) ISBN: (pdf) 1. Mathematics Examinations, questions, etc. 2. Educational tests and measurements Manitoba. 3. Mathematics Study and teaching (Secondary) Manitoba. 4. Pre-calculus Study and teaching (Secondary) Manitoba. 5. Mathematical ability Testing. I. Manitoba. Manitoba Education and Training Manitoba Education and Training Winnipeg, Manitoba, Canada All exemplars found in this resource are copyright protected and should not be extracted, accessed, or reproduced for any purpose other than for their intended educational use in this resource. Sincere thanks to the students who allowed their original material to be used. Permission is hereby given to reproduce this resource for non-profit educational purposes provided the source is cited. After the administration of this test, print copies of this resource will be available for purchase from the Manitoba Learning Resource Centre. Order online at This resource will also be available on the Manitoba Education and Training website at Websites are subject to change without notice. Disponible en français. While the department is committed to making its publications as accessible as possible, some parts of this document are not fully accessible at this time. Available in alternate formats upon request.

3 Table of Contents General Marking Instructions... 1 Scoring Guidelines for Booklet 1 Questions... 5 Scoring Guidelines for Booklet 2 Questions Answer Key for Selected Response Questions Appendices Appendix A: Marking Guidelines Appendix B: Irregularities in Provincial Tests Irregular Test Booklet Report Appendix C: Table of Questions by Unit and Learning Outcome i

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5 General Marking Instructions Please do not make any marks in the student test booklets. If the booklets have marks in them, the marks will need to be removed by departmental staff prior to sample marking should the booklet be selected. Please ensure that the booklet number and the number on the Answer/Scoring Sheet are identical students and markers use only a pencil to complete the Answer/Scoring Sheets the totals of each of the four parts are written at the bottom each student s final result is recorded, by booklet number, on the corresponding Answer/Scoring Sheet the Answer/Scoring Sheet is complete a photocopy has been made for school records Once marking is completed, please forward the Answer/Scoring Sheets to Manitoba Education and Training in the envelope provided (for more information see the administration manual). Marking the Test Questions The test is composed of constructed response questions and selected response questions. Constructed response questions are worth 1 to 5 marks each, and selected response questions are worth 1 mark each. An answer key for the selected response questions can be found at the beginning of the section Booklet 2 Questions. To receive full marks, a student s response must be complete and correct. Where alternative answering methods are possible, the Marking Guide attempts to address the most common solutions. For general guidelines regarding the scoring of students responses, see Appendix A. Irregularities in Provincial Tests During the administration of provincial tests, supervising teachers may encounter irregularities. Markers may also encounter irregularities during local marking sessions. Appendix B provides examples of such irregularities as well as procedures to follow to report irregularities. If an Answer/Scoring Sheet is marked with 0 and/or NR only (e.g., student was present but did not attempt any questions), please document this on the Irregular Test Booklet Report. Pre-Calculus Mathematics: Marking Guide (June 2017) 1

6 Assistance If, during marking, any marking issue arises that cannot be resolved locally, please call Manitoba Education and Training at the earliest opportunity to advise us of the situation and seek assistance if necessary. You must contact the Assessment Consultant responsible for this project before making any modifications to the answer keys or scoring rubrics. Youyi Sun Assessment Consultant Grade 12 Pre-Calculus Mathematics Telephone: Toll-Free: , extension Pre-Calculus Mathematics: Marking Guide (June 2017)

7 Communication Errors The marks allocated to questions are primarily based on the concepts and procedures associated with the learning outcomes in the curriculum. For each question, shade in the circle on the Answer/Scoring Sheet that represents the marks given based on the concepts and procedures. A total of these marks will provide the preliminary mark. Errors that are not related to concepts or procedures are called Communication Errors (see Appendix A) and will be tracked on the Answer/Scoring Sheet in a separate section. There is a ½ mark deduction for each type of communication error committed, regardless of the number of errors per type (i.e., committing a second error for any type will not further affect a student s mark), with a maximum deduction of 5 marks from the total test mark. When a given response includes multiple types of communication errors, deductions are indicated in the order in which the errors occur in the response. No communication errors are recorded for work that has not been awarded marks. The total deduction may not exceed the marks awarded. The student s final mark is determined by subtracting the communication errors from the preliminary mark. Example: A student has a preliminary mark of 72. The student committed two E1 errors (½ mark deduction), four E7 errors (½ mark deduction), and one E8 error (½ mark deduction). Although seven communication errors were committed in total, there is a deduction of only 1½ marks. COMMUNICATION ERRORS / ERREURS DE COMMUNICATION Shade in the circles below for a maximum total deduction of 5 marks (0.5 mark deduction per error). Noircir les cercles ci-dessous pour une déduction maximale totale de 5 points (déduction de 0,5 point par erreur). E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 Example: Marks assigned to the student. Marks Awarded Booklet 1 25 Selected Response 7 Booklet 2 Total Marks Communication Errors (Deduct) 1½ maximum deduction of 5 marks Total 70½ 90 Pre-Calculus Mathematics: Marking Guide (June 2017) 3

8 4 Pre-Calculus Mathematics: Marking Guide (June 2017)

9 Scoring Guidelines for Booklet 1 Questions Pre-Calculus Mathematics: Marking Guide (June 2017) 5

10 Question 1 T1 A section of a car windshield is cleaned by a wiper as shown in the diagram below. The arm of the wiper is 22 inches, and it rotates through a central angle of 132. Determine the length of the arc that is created by the tip of the wiper. Solution in. π θ = mark for conversion 132π 11π = or s = θ r 11π s = (22) π s = inches 15 1 mark for substitution 2 marks or s = inches 6 Pre-Calculus Mathematics: Marking Guide (June 2017)

11 Exemplar 1 1½ out of 2 award full marks ½ mark for procedural error of incorrect radius E5 (units of measure omitted in final answer) Exemplar 2 2 out of 2 award full marks E6 (rounding too early) Exemplar 3 1 out of mark for substitution E5 (units of measure omitted in final answer) Pre-Calculus Mathematics: Marking Guide (June 2017) 7

12 Question 2 P3 There are 20 boys and 11 girls who can be selected to be on a team. Determine the number of ways that 7 boys and 5 girls can be selected for this team. Solution C C = ½ mark for 20 7 C ½ mark for 11 5 C 1 mark for the product of combinations 2 marks 8 Pre-Calculus Mathematics: Marking Guide (June 2017)

13 Exemplar 1 1 out of 2 award full marks 1 mark for concept error (permutations instead of combinations) Exemplar 2 1½ out of 2 award full marks ½ mark for arithmetic error in line 4 Exemplar 3 1 out of 2 + ½ mark for C ½ mark for C 11 5 Pre-Calculus Mathematics: Marking Guide (June 2017) 9

14 Question 3 R10 A water filtration system which removes impurities from a sample of water can be modelled by n P = , where: ( ) P = the percentage of impurities remaining, in decimal form n = the number of filters Determine, algebraically, how many filters are required so that less than 1% of the impurities remain in the water sample. Express your answer as a whole number. Solution ( ) 0.04 = ( 0.55) ( ) = n ( ) ( ) = n ( ) log 0.04 log 0.55 log 0.04 log 0.55 n 0.01 = n n = filters are needed ½ mark for substitution ½ mark for applying logarithms ½ mark for power law ½ mark for solving for n 2 marks 10 Pre-Calculus Mathematics: Marking Guide (June 2017)

15 Exemplar 1 1½ out of 2 + ½ mark for applying logarithms + ½ mark for power law + ½ mark for solving for n E6 rounding error (answer not expressed as a whole number) Pre-Calculus Mathematics: Marking Guide (June 2017) 11

16 Exemplar 2 1 out of 2 award full marks 1 mark for concept error in line 3 E6 rounding error (not rounding up) 12 Pre-Calculus Mathematics: Marking Guide (June 2017)

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18 Question 4 P4 In the binomial expansion of Solution x y, determine the middle term in simplified form. 2 ( ) t = C x y 8 16 t = 70x 5 4 y x t = 5 4 y 2 marks (1 mark for C, ½ mark for each consistent factor) mark for simplification (½ mark for coefficient, ½ mark for exponents) 3 marks 14 Pre-Calculus Mathematics: Marking Guide (June 2017)

19 Exemplar 1 2 out of mark for 8 4 C + 1 mark for consistent factors Pre-Calculus Mathematics: Marking Guide (June 2017) 15

20 Exemplar 2 2 out of mark for consistent factors + 1 mark for simplification 16 Pre-Calculus Mathematics: Marking Guide (June 2017)

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22 Question 5 T5 Solve the following equation algebraically over the interval [ 0, 2 π ]. Solution 2 6sin θ + sinθ 1 = 0 ( θ ) ( θ ) 3sin 1 2sin + 1 = 0 3sinθ 1 = 0 2sinθ + 1 = sinθ = sinθ = 3 2 θ = r 7π θ = θ = 6 11π θ = θ = 6 or θ = 0.340, 2.802, 3.665, mark for solving for sinθ (½ mark for each branch) 2 marks (½ mark for each value of θ ) 3 marks 18 Pre-Calculus Mathematics: Marking Guide (June 2017)

23 Exemplar 1 1½ out of mark for solving for sinθ + 1 mark for consistent values of θ ½ mark for arithmetic error in line 5 E2 (changing an equation to an expression in lines 2 to 4) E3 (variable omitted in line 3) E5 (answer stated in degrees instead of radians) Pre-Calculus Mathematics: Marking Guide (June 2017) 19

24 Exemplar 2 2½ out of 3 award full marks ½ mark for arithmetic error in line 2 20 Pre-Calculus Mathematics: Marking Guide (June 2017)

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26 Question 6 R4, R5 Given the graph of y = f ( x), sketch the graph of y f ( x 4) y = f ( x) x Solution y 1 mark for reflection over the y-axis 1 mark for horizontal translation 1 1 x 2 marks 22 Pre-Calculus Mathematics: Marking Guide (June 2017)

27 Exemplar 1 1 out of mark for reflection over the y-axis Exemplar 2 1 out of mark for reflection over the y-axis Pre-Calculus Mathematics: Marking Guide (June 2017) 23

28 Exemplar 3 1 out of mark for horizontal translation 24 Pre-Calculus Mathematics: Marking Guide (June 2017)

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30 Question 7 T2, T3 Given the following triangle, determine csc θ. y 2 θ x 5 Solution x + y = r y = 5 2 y = 21 y = ± 21 ½ mark for substitution ½ mark for solving for y cscθ = mark for cscθ (½ mark for quadrant, ½ mark for value) 2 marks Note(s): Accept any of the following values for y: y = ± 21, y = 21, or y = Pre-Calculus Mathematics: Marking Guide (June 2017)

31 Exemplar 1 1½ out of 2 + ½ mark for substitution + ½ mark for solving for y + ½ mark for value of cscθ Pre-Calculus Mathematics: Marking Guide (June 2017) 27

32 Exemplar 2 1½ out of 2 + ½ mark for substitution + ½ mark for solving for y + ½ mark for quadrant of cscθ 28 Pre-Calculus Mathematics: Marking Guide (June 2017)

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34 Question 8 P2 Solve algebraically. n P 2 = 9 n Solution n! ( n 2! ) ( 1) ( n 2! ) ( n 2! ) n( n ) n n 2 n n = 9n 2 n 10n = 0 ( ) = 9n = 9n 1 = 9n n n 10 = 0 n = 0 n = 10 ½ mark for substitution into equation 1 mark for factorial expansion ½ mark for simplification of factorials ½ mark for rejecting the extraneous root ½ mark for the value of n 3 marks 30 Pre-Calculus Mathematics: Marking Guide (June 2017)

35 Exemplar 1 2½ out of 3 + ½ mark for substitution + 1 mark for factorial expansion + ½ mark for simplification of factorials + ½ mark for the value of n Pre-Calculus Mathematics: Marking Guide (June 2017) 31

36 Exemplar 2 2 out of 3 + ½ mark for substitution + 1 mark for factorial expansion + ½ mark for simplification of factorials 32 Pre-Calculus Mathematics: Marking Guide (June 2017)

37 Exemplar 3 3 out of 3 Pre-Calculus Mathematics: Marking Guide (June 2017) 33

38 Question 9 R13 Describe the transformations applied to the graph of f ( x ) to obtain the graph of g( x ). y 1 1 f x ( x) g( x) Solution A reflection over the x-axis then a vertical translation of one unit down. or A vertical translation of one unit up then a reflection over the x-axis. 1 mark for reflection over the x-axis 1 mark for vertical translation 2 marks Note(s): Award a maximum of 1 mark if correct transformations are given in the incorrect order. 34 Pre-Calculus Mathematics: Marking Guide (June 2017)

39 Exemplar 1 0 out of 2 Exemplar 2 1 out of 2 award full marks 1 mark for concept error (incorrect order) Exemplar 3 ½ out of mark for vertical translation ½ mark for lack of clarity in description Pre-Calculus Mathematics: Marking Guide (June 2017) 35

40 Question 10 R12 Determine, algebraically, the value of the leading coefficient of the graph of the polynomial function, p( x ). y p( x) x 12 Solution ( ) = ( + 3) ( + 1) ( 1) p x a x x x 3 ( ) ( ) ( ) 12 = a = 3a a = 4 3 ½ mark for factors of px ( ) ½ mark for odd multiplicity for ( x + 1) greater than 1 ½ mark for substituting p ( 0) = 12 ½ mark for value of a 2 marks 36 Pre-Calculus Mathematics: Marking Guide (June 2017)

41 Exemplar 1 1½ out of 2 + ½ mark for factors of p( x ) + ½ mark for substituting p ( 0) = 12 + ½ mark for value of a Exemplar 2 ½ out of 2 + ½ mark for factors of p( x ) Pre-Calculus Mathematics: Marking Guide (June 2017) 37

42 Exemplar 3 1½ out of 2 + ½ mark for factors of p( x ) + ½ mark for odd multiplicity for ( x + 1) greater than 1 + ½ mark for value of a Exemplar 4 1 out of 2 + ½ mark for factors of p( x ) + ½ mark for odd multiplicity for ( x + 1) greater than 1 38 Pre-Calculus Mathematics: Marking Guide (June 2017)

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44 Question 11 P1 Frank, Liam, Chan, and Thao are going to a movie. Determine the number of ways they can sit in a row of four chairs, if Frank and Chan must sit beside each other. Solution 3!2! 12 ½ mark for 3! ½ mark for 2! as a product of factorials 1 mark 40 Pre-Calculus Mathematics: Marking Guide (June 2017)

45 Exemplar 1 0 out of 1 Exemplar 2 ½ out of 1 award full marks ½ mark for arithmetic error Exemplar 3 1 out of 1 award full marks E1 (final answer not stated) Exemplar 4 ½ out of 1 + ½ mark for 3! Pre-Calculus Mathematics: Marking Guide (June 2017) 41

46 Question 12 R6 Determine, algebraically, if f ( x) = 1 x + 5 and g( x) = 1 x 5 are inverses of each other. Justify your answer. Solution Method 1 Let f x = y f 1 ( ) 1 y = x x = y y + 5 = x 1 y = 5 x 1 ( x) = 5 x 1 ( ) ( ) f x g x 1 mark for switching x and y values ½ mark for solving for y ½ mark for justification 2 marks Method 2 1 f ( g( x) ) = x 5 1 = 1 + 5x 25 x 5 1 = 5x 24 x 5 x 5 = 5x 24 1 mark for ( ) ( ) or ( ( )) f g x g f x ½ mark for simplification ( ( )) g f x 1 = 1 5 x = 1 5x 25 x = 5x 24 x + 5 x + 5 = 5x 24 ( ( )) ( ) ( ) They are not inverses because f g x x or g f x x. ½ mark for justification 2 marks 42 Pre-Calculus Mathematics: Marking Guide (June 2017)

47 Exemplar 1 1 out of mark for f ( g( x) ) or g( f ( x )) Exemplar 2 1½ out of 2 award full marks ½ mark for arithmetic error in line 6 E4 (missing brackets but still implied in line 4) Pre-Calculus Mathematics: Marking Guide (June 2017) 43

48 Question 13 R13 Using the graphs of y = f ( x) and y = g( x), solve f ( x) = g( x). y g( x) f ( x) 1 1 x Solution x = 2 1 mark 44 Pre-Calculus Mathematics: Marking Guide (June 2017)

49 Exemplar 1 0 out of 1 award full marks 1 mark for concept error (including the y-value) Exemplar 2 0 out of 1 award full marks 1 mark for concept error (including the y-value ) Pre-Calculus Mathematics: Marking Guide (June 2017) 45

50 Question 14 T1 An angle in standard position measures 3 π. 4 Determine in which quadrant the terminal arm of this angle is located after a rotation of 3 radians. Justify your answer. Solution 3π radians Quadrant IV or The angle 3 π terminates in quadrant II. A rotation of 3 radians is almost a half rotation; 4 therefore, the terminal arm is located in quadrant IV. 1 mark for justification 1 mark 46 Pre-Calculus Mathematics: Marking Guide (June 2017)

51 Exemplar 1 1 out of 1 award full marks E1 (final answer not stated) Exemplar 2 0 out of 1 Exemplar 3 0 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 47

52 Question 15 T6 Prove the following identity for all permissible values of θ. sin 2θ cotθ 1 cos 2θ = Solution Left-Hand Side Right-Hand Side sin 2θ 1 cos 2θ 2 sinθ cosθ 2 ( θ ) 1 1 2sin θ 2 sin cos 2 2sin cosθ sinθ cotθ θ θ cotθ 1 mark for correct substitution of identities 1 mark for appropriate algebraic strategies 1 mark for logical process to prove the identity 3 marks 48 Pre-Calculus Mathematics: Marking Guide (June 2017)

53 Exemplar 1 2 out of mark for correct substitution of identities + 1 mark for logical process to prove the identity Pre-Calculus Mathematics: Marking Guide (June 2017) 49

54 Exemplar 2 1 out of mark for correct substitution of identities 50 Pre-Calculus Mathematics: Marking Guide (June 2017)

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56 Question 16 R3 If the range of y f ( x) = is 3 y 6, determine the range of y = f ( x) 2 3. Solution [ 6,12] or 6 y 12 1 mark 52 Pre-Calculus Mathematics: Marking Guide (June 2017)

57 Exemplar 1 1 out of 1 award full marks E8 (bracket error made when stating range) Exemplar 2 0 out of 1 Exemplar 3 0 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 53

58 Question 17 T5 Maurice incorrectly solved the equation, sinθ + 1 = 0, over the interval 0, 360. Describe his error. Solution Maurice should have written that θ is equal to 270 not sinθ = mark 54 Pre-Calculus Mathematics: Marking Guide (June 2017)

59 Exemplar 1 0 out of 1 Exemplar 2 0 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 55

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61 Scoring Guidelines for Booklet 2 Questions Pre-Calculus Mathematics: Marking Guide (June 2017) 57

62 Answer Key for Selected Response Questions Question Answer Learning Outcome 18 D R2 19 B R5 20 B R7 21 C T4 22 B R9 23 D R12 24 D T1 25 C P3 26 A R14 27 B, C, A, D R13 58 Pre-Calculus Mathematics: Marking Guide (June 2017)

63 Question 18 R2 If P ( 3, 5) is a point on the graph of y f ( x) y = f ( x 1) + 7. a) ( 2,12 ) b) ( 4, 2) c) ( 2, 2) d) ( 4,12 ) =, identify the corresponding point on the graph of Question 19 R5 Identify how the graph of y = 3 x x is transformed to the graph of y = 3. a) reflected over the x-axis b) reflected over the y-axis c) reflected over both the x-axis and the y-axis d) reflected over the line y = x Question 20 R7 Identify the equation log a b = c in exponential form. a) b) c) c b c a b a = a = b = c d) a c = b Pre-Calculus Mathematics: Marking Guide (June 2017) 59

64 Question 21 T4 Identify the graph of y = tan x. a) b) y y 1 π 2 x 11 π 2 x c) d) y y 1 π 2 x 1 π 2 x 60 Pre-Calculus Mathematics: Marking Guide (June 2017)

65 Question 22 R9 Identify which of the following graphs represents a logarithmic function. a) b) y y 1 1 x 1 1 x c) d) y y 1 1 x 1 1 x Pre-Calculus Mathematics: Marking Guide (June 2017) 61

66 Question 23 R12 If the volume of a box is represented by V ( x) ( x ) ( x ) ( x ) value of x. a) 4 b) 1 c) 1 d) 4 = , identify a possible Question 24 T1 π Identify a coterminal angle for θ =. 3 a) b) c) π 3 4π 3 7π 3 d) 11 π 3 Question 25 P3 Identify the value of n in the equation C = C n 3 n 6. a) 3 b) 6 c) 9 d) Pre-Calculus Mathematics: Marking Guide (June 2017)

67 Question 26 R14 Identify the equation of the function, f ( x ), for the following graph. y f ( x) 1 1 x a) f ( x) b) f ( x) c) f ( x) d) f ( x) = = = = 2x x x + 3 2x x x 2 ( + 3) 3x x x 2 ( + 2) Pre-Calculus Mathematics: Marking Guide (June 2017) 63

68 Question 27 R13 Match the following radical functions with their graphs. Solution Place the appropriate letter in this column. ( ) = 2 ( + 3) f x x ( ) = 2 ( x+ 3) g x ( ) = 3 ( x 2) h x B C A ½ mark for each correct answer 2 marks ( ) = 3( x 2) k x A) y B) D y 1 1 x 1 1 x C) y D) y 1 1 x 1 1 x 64 Pre-Calculus Mathematics: Marking Guide (June 2017)

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70 Question 28 R11 Express ( ) 3 2 = as a product of factors. p x x x x Solution p 3 ( ) ( ) ( ) 2 = = = 0 1 mark for identifying one possible value of x ( ) = ( 2) ( 4) p x x x or ( ) = ( 2) ( 2) ( + 2) p x x x x 1 mark for synthetic division (or any equivalent strategy) 1 mark for product of factors 3 marks or ( ) = ( 2) ( + 2) 2 p x x x 66 Pre-Calculus Mathematics: Marking Guide (June 2017)

71 Exemplar 1 2½ out of 3 award full marks ½ mark for procedural error (solving for the roots) E2 (changing an equation to an expression in line 6) Exemplar 2 2 out of mark for identifying one possible value of x + 1 mark for synthetic division E7 (notation error in lines 2 to 4) Pre-Calculus Mathematics: Marking Guide (June 2017) 67

72 Question 29 R1 Given the graph of f ( x) = ( x+ ) ( x ) 3 1, y f ( x) 1 1 x a) sketch the graph of g( x) =. f ( x) b) describe how to sketch the graph of h( x) = f ( x). 1 Solution a) 1 y 1 f ( x) x 1 mark for asymptotic behaviour at x = 1 and x = 3 ½ mark for asymptotic behaviour at y = 0 ½ mark for graph left of x = 3 ½ mark for graph between x = 3 and x = 1 ½ mark for graph right of x = 1 3 marks b) Change all negative y-values to positive values. The positive y-values do not change. Note(s): 1 mark Award 1 mark if only vertical asymptotes at x = 1 and x = 3 are drawn. 68 Pre-Calculus Mathematics: Marking Guide (June 2017)

73 Exemplar 1 a) 1½ out of mark for asymptotic behaviour at x = 3 and x = 1 + ½ mark for graph right of x = 1 b) 1 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 69

74 Exemplar 2 a) 2½ out of mark for asymptotic behaviour at x = 3 and x = 1 + ½ mark for asymptotic behaviour at y = 0 + ½ mark for graph left of x = 3 + ½ mark for graph right of x = 1 E10 (asymptotes omitted but still implied) b) ½ out of 1 award full marks ½ mark for lack of clarity in description 70 Pre-Calculus Mathematics: Marking Guide (June 2017)

75 Exemplar 3 a) 2½ out of 3 award full marks ½ mark for procedural error (correct shape with no correct points) b) 1 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 71

76 Exemplar 4 a) 1 out of mark for vertical asymptotes at x = 3 and x = 1 (see note) b) ½ out of 1 award full marks ½ mark for lack of clarity in description 72 Pre-Calculus Mathematics: Marking Guide (June 2017)

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78 Question 30 R9 Describe how the value of m in the equation ( ) on the graph of y = log x. 3 Solution y log x m, m 3 =, affects the asymptote The vertical asymptote is translated horizontally left or right m units from the y-axis. 1 mark 74 Pre-Calculus Mathematics: Marking Guide (June 2017)

79 Exemplar 1 ½ out of 1 award full marks ½ mark for lack of clarity in description Exemplar 2 ½ out of 1 award full marks ½ mark for lack of clarity in description Exemplar 3 0 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 75

80 Question 31 R10 Solve algebraically. x 25 3x+ 1 1 = 5 Solution ( 2 x ) ( ) 3 x+ = 1 2x 3x 1 5 = 5 2x = 3x 1 x = 1 1 mark for changing to a common base ½ mark for exponent law ½ mark for equating exponents 2 marks 76 Pre-Calculus Mathematics: Marking Guide (June 2017)

81 Exemplar 1 ½ out of 2 + ½ mark for exponent law E7 (transcription error in line 1) Exemplar 2 1½ out of 2 award full marks ½ mark for arithmetic error in line 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 77

82 Exemplar 3 2 out of 2 award full marks E1 (final answer not stated) 78 Pre-Calculus Mathematics: Marking Guide (June 2017)

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84 Question 32 T5, T6 Solve cos 2θ = 0, where θ. Solution Method sin θ = 0 sin 2 1 θ = sinθ = sinθ = 2 2 π 3π 5π 7π θ =, θ =, π θ = +πk 4 π πk where k or θ = +, where k 3π 4 2 = +πk 4 1 mark for double-angle identity 1 mark for solving for sinθ 1 mark for values of θ (½ mark for each branch) 1 mark for general solution 4 marks Method cos θ 1 = 0 cos 2 1 θ = cosθ = cosθ = 2 2 π 7π 3π 5π θ =, θ =, mark for double-angle identity 1 mark for solving for cosθ 1 mark for values of θ (½ mark for each branch) π θ = +πk 4 π πk where k or θ = +, where k 1 mark for general solution 3π 4 2 = +πk 4 4 marks Note(s): Deduct a maximum of 1 mark if student omits second branch when taking the square root. 80 Pre-Calculus Mathematics: Marking Guide (June 2017)

85 Question 32 T5, T6 Method 3 cos 2θ = 0 π 3π 2 θ =, 2 2 π 2θ = + 2πk 2 where 3π 2θ = + 2πk 2 π θ = +πk 4 3π θ = +πk 4 k 2 marks for values of 2θ (1 mark for each value) 1 mark for general solution 1 mark for values of θ 4 marks Pre-Calculus Mathematics: Marking Guide (June 2017) 81

86 Exemplar 1 3 out of mark for double-angle identity + 1 mark for values of θ + 1 mark for general solution Exemplar 2 2 out of marks for values of 2θ + 1 mark for values of θ 1 mark for concept error in lines 3 to 5 82 Pre-Calculus Mathematics: Marking Guide (June 2017)

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88 Question 33 R12 Describe a difference between the graphs of y f ( x) Solution ( ) = 2( + 1) ( + 3) f x x x ( ) = 2( + 1) ( + 3) 2 2 g x x x = and y = g( x). The end behaviour is different, because the graph of f ( x ) falls to the right and the graph of g( x ) rises to the right. or The y-intercept of f ( x ) is negative while the y-intercept of g( x ) is positive. or One graph is a reflection over the x-axis of the other graph. 1 mark 84 Pre-Calculus Mathematics: Marking Guide (June 2017)

89 Exemplar 1 ½ out of 1 award full marks ½ mark for lack of clarity in description Exemplar 2 0 out of 1 Exemplar 3 0 out of 1 Exemplar 4 1 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 85

90 Question 34 R13 Given the graph of y = f ( x), sketch the graph of f ( x ). y 1 1 x f ( x) Solution y 1 mark for restricted domain ½ mark for shape between both invariant points, 0 y 1 { } ½ mark for shape above invariant points, { 1 y 2} 1 1 x 2 marks f ( x) 86 Pre-Calculus Mathematics: Marking Guide (June 2017)

91 Exemplar 1 f ( x) 1½ out of mark for restricted domain + ½ mark for shape above invariant points, { 1 y 2} Exemplar 2 f ( x) 1½ out of mark for restricted domain + ½ mark for shape above invariant points, { 1 y 2} Pre-Calculus Mathematics: Marking Guide (June 2017) 87

92 Question 35 R12 Describe the relationship between the zeros of the function ( ) ( ) ( ) 2 roots of the equation ( ) ( ) 2 f x = 2x 1 x+ 3, the 2x 1 x+ 3 = 0, and the x -intercepts of the graph of y = f ( x). Solution The zeros, roots, and x -intercepts all have the same values. 1 mark 88 Pre-Calculus Mathematics: Marking Guide (June 2017)

93 Exemplar 1 ½ out of 1 award full mark ½ mark for lack of clarity in description Exemplar 2 ½ out of 1 award full marks ½ mark for lack of clarity in description Exemplar 3 0 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 89

94 Question 36 T4 Sketch a graph of at least one period of the function ( ) Solution y 1 π f x = cos x π 1 π 2π 3π 4π x 1 mark for period 1 mark for horizontal translation 1 mark for vertical translation 2 3 marks Pre-Calculus Mathematics: Marking Guide (June 2017)

95 Exemplar 1 2 out of mark for horizontal translation + 1 mark for vertical translation Exemplar 2 2½ out of 3 award full marks ½ mark for incorrect shape of graph E9 (arrowheads incorrect) Pre-Calculus Mathematics: Marking Guide (June 2017) 91

96 Exemplar 3 2 out of mark for period + 1 mark for horizontal translation 92 Pre-Calculus Mathematics: Marking Guide (June 2017)

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98 Question 37 T6 4π Verify that θ = is a solution of the equation cos 1 0. θ = Solution 2 4π Left-hand side = 4 cos = = = 0 = Right-hand side 4π ½ mark for value of cos 3 ½ mark for verification 1 mark 94 Pre-Calculus Mathematics: Marking Guide (June 2017)

99 Exemplar 1 ½ out of 1 + ½ mark for verification E7 (transcription error in line 2) Pre-Calculus Mathematics: Marking Guide (June 2017) 95

100 Exemplar 2 ½ out of 1 + ½ mark for verification E7 (notation error on lines 2 to 5) 96 Pre-Calculus Mathematics: Marking Guide (June 2017)

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102 Question 38 R14 Describe how to determine the equation of the horizontal asymptote of a rational function when the degree of the polynomial in the numerator and the degree of the polynomial in the denominator are equal. Solution a The horizontal asymptote is y = where a is the leading coefficient in the numerator and b b is the leading coefficient in the denominator. or Use polynomial division to divide the numerator by the denominator. The equation of the horizontal asymptote will have the same value as the quotient. 1 mark 98 Pre-Calculus Mathematics: Marking Guide (June 2017)

103 Exemplar 1 0 out of 1 Exemplar 2 ½ out of 1 award full marks ½ mark for lack of clarity in description Exemplar 3 ½ out of 1 award full marks ½ mark for lack of clarity in description Exemplar 4 0 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 99

104 Question 39 T3 Evaluate. 5π cot 6 17π sin 3 Solution ( 3) π 1 mark for cot (½ mark for value, ½ mark for quadrant) 6 17π 1 mark for sin (½ mark for value, ½ mark for quadrant) 3 2 marks 100 Pre-Calculus Mathematics: Marking Guide (June 2017)

105 Exemplar 1 1½ out of 2 π mark for cot 6 17π + ½ mark for value of sin 3 Exemplar 2 1½ out of 2 π mark for cot 6 + ½ mark for quadrant of 17π sin 3 Pre-Calculus Mathematics: Marking Guide (June 2017) 101

106 Question 40 R14 Sketch the graph of the function f ( x) = 1 ( x 1) 2 and determine the range. Solution y 1 mark for asymptotic behaviour at x = 1 1 mark for asymptotic behaviour at y = 0 ½ mark for graph left of x = 1 ½ mark for graph right of x = x Range: { y y < 0} or,0 Range: ( ) 1 mark for range (consistent with graph) 4 marks 102 Pre-Calculus Mathematics: Marking Guide (June 2017)

107 Exemplar 1 3 out of mark for asymptotic behaviour at 0 y = + ½ mark for graph left of asymptote + ½ mark for graph right of asymptote + 1 mark for range E8 (bracket error made when stating range) E10 (asymptotes omitted but still implied) Pre-Calculus Mathematics: Marking Guide (June 2017) 103

108 Exemplar 2 3½ out of mark for asymptotic behaviour at x = mark for asymptotic behaviour at y = 0 + ½ mark for graph left of x = mark for range (consistent with graph) E9 (scale values on axes not indicated) 104 Pre-Calculus Mathematics: Marking Guide (June 2017)

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110 Question 41 R1 Given f ( x) = x 2 and g( x) a) determine g( f ( x )). 2 = x + 1, b) explain why the domain of g( f ( x )) is restricted. Solution a) ( ( )) ( ) 2 g f x = x = x 2 + 1, x 2 = x 1, x 2 1 mark for composition 1 mark b) The domain of g( f ( x )) must be restricted because the domain of f ( ) x is restricted. or The value of the radicand must be positive. 1 mark 106 Pre-Calculus Mathematics: Marking Guide (June 2017)

111 Exemplar 1 a) g( f( x )) = 1 out of 1 award full marks E7 (notation error in line 1) b) 1 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 107

112 Exemplar 2 a) ½ out of 1 award full marks ½ mark for procedural error (solving for x) b) 1 out of Pre-Calculus Mathematics: Marking Guide (June 2017)

113 Exemplar 3 a) 1 out of 1 b) ½ out of 1 award full marks ½ mark for terminology error in explanation Pre-Calculus Mathematics: Marking Guide (June 2017) 109

114 Question 42 R7, R10 Solve algebraically. 2 log 3 + log 4 = 2, where a > 0 a a Solution 2 ( ) log 3 4 = 2 a log 36 = 2 a 2 a = 36 a = 6 1 mark for power law 1 mark for product law 1 mark for exponential form 3 marks 110 Pre-Calculus Mathematics: Marking Guide (June 2017)

115 Exemplar 1 2 out of mark for product law + 1 mark for exponential form Exemplar 2 2½ out of 3 award full marks ½ mark for arithmetic error in line 2 Pre-Calculus Mathematics: Marking Guide (June 2017) 111

116 Exemplar 3 3 out of 3 award full marks E8 (answer outside the given domain) 112 Pre-Calculus Mathematics: Marking Guide (June 2017)

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118 Question 43 T5 Solve sec 2 0 Solution θ + = over the interval [ π ] 0, 2. secθ + 2 = 0 secθ = 2 1 cosθ = 2 2π 4π θ =, mark for reciprocal 1 mark for values of θ (½ mark for each value) 2 marks 114 Pre-Calculus Mathematics: Marking Guide (June 2017)

119 Exemplar 1 1½ out of mark for reciprocal 7π 5π + ½ mark for consistent value of θ ( is consistent with the reference angle of ) 6 6 Exemplar 2 1 out of mark for reciprocal Exemplar 3 0 out of 2 Pre-Calculus Mathematics: Marking Guide (June 2017) 115

120 Question 44 R9 Determine the x-intercept of the graph of f ( x) = e 1. x Solution Method 1 x 0 = e 1 x 1 = e x = 0 ½ mark for substitution ½ mark for solving for x 1 mark Method 2 y 1 1 x x = 0 1 mark 116 Pre-Calculus Mathematics: Marking Guide (June 2017)

121 Exemplar 1 1 out of 1 award full marks E1 (final answer not stated) Exemplar 2 0 out of 1 Exemplar 3 ½ out of 1 + ½ mark for substitution Pre-Calculus Mathematics: Marking Guide (June 2017) 117

122 Question 45 P4 Given the 5 th row of Pascal s triangle, determine the values of the next row Solution mark 118 Pre-Calculus Mathematics: Marking Guide (June 2017)

123 Exemplar 1 0 out of 1 Exemplar 2 0 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 119

124 Question 46 R7, R8 Evaluate. log 80 log Solution 80 log 2 10 log mark for quotient law 1 mark for evaluating the logarithm 2 marks 120 Pre-Calculus Mathematics: Marking Guide (June 2017)

125 Exemplar 1 1 out of mark for quotient law Exemplar 2 2 out of 2 award full marks E3 (variables introduced without being defined in line 3) Pre-Calculus Mathematics: Marking Guide (June 2017) 121

126 Question 47 T4 State the amplitude of f ( x) ( x ) = 2sin π 1. Solution 2 1 mark 122 Pre-Calculus Mathematics: Marking Guide (June 2017)

127 Exemplar 1 0 out of 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 123

128 Question 48 T3, T6 Determine the exact value of cos15. Solution ( ) = ( ) cos 15 cos = cos 60 cos 45 + sin 60 sin = = = 4 1 mark for substitution into correct identity 2 marks for values (½ mark for each exact value) 3 marks Note(s): Other combinations are possible. 124 Pre-Calculus Mathematics: Marking Guide (June 2017)

129 Exemplar 1 1 out of 3 + ½ mark for value of cos 60 + ½ mark for value of cos 45 Exemplar 2 2½ out of 3 award full marks ½ mark for arithmetic error in line 3 E4 (missing brackets but still implied in line 1) Pre-Calculus Mathematics: Marking Guide (June 2017) 125

130 Exemplar 3 1 out of marks for values ½ mark for procedural error in line 1 ½ mark for arithmetic error in line 4 E2 (changing an equation to an expression in lines 1 and 3) 126 Pre-Calculus Mathematics: Marking Guide (June 2017)

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132 Question 49 R1 2 Given f ( x) = x + 5x+ 6, g( x) = x+ 3, and h( x) = f ( x) g( x), a) determine h( x ). b) sketch the graph of y = h( x). Solution a) 2 f x g x x x x ( ) ( ) = ( + 3) 2 = x + 4x+ 3 1 mark b) y 1 mark for graph consistent with a) 1 1 x 1 mark 128 Pre-Calculus Mathematics: Marking Guide (June 2017)

133 Exemplar 1 a) ½ out of 1 award full marks ½ mark for arithmetic error E7 (notation error in line 1) b) 1 out of 1 graph consistent with answer in a) Pre-Calculus Mathematics: Marking Guide (June 2017) 129

134 Exemplar 2 a) 0 out of 1 b) 1 out of 1 graph consistent with error in a) 130 Pre-Calculus Mathematics: Marking Guide (June 2017)

135 Appendices Pre-Calculus Mathematics: Marking Guide (June 2017) 131

136 132 Pre-Calculus Mathematics: Marking Guide (June 2017)

137 Appendix A MARKING GUIDELINES Errors that are conceptually related to the learning outcomes associated with the question will result in a 1 mark deduction. Each time a student makes one of the following errors, a ½ mark deduction will apply. arithmetic error procedural error terminology error in explanation lack of clarity in explanation, description, or justification incorrect shape of graph (only when marks are not allocated for shape) Communication Errors The following errors, which are not conceptually related to the learning outcomes associated with the question, may result in a ½ mark deduction and will be tracked on the Answer/Scoring Sheet. E1 final answer E2 equation/expression E3 variables E4 brackets E5 units E6 rounding E7 notation/transcription E8 domain/range E9 graphing E10 asymptotes answer given as a complex fraction final answer not stated changing an equation to an expression or vice versa equating the two sides when proving an identity variable omitted in an equation or identity variables introduced without being defined 2 2 sin x written instead of sin x missing brackets but still implied units of measure omitted in final answer incorrect units of measure answer stated in degrees instead of radians or vice versa rounding error rounding too early notation error transcription error answer outside the given domain bracket error made when stating domain or range domain or range written in incorrect order endpoints or arrowheads omitted or incorrect scale values on axes not indicated coordinate points labelled incorrectly asymptotes drawn as solid lines asymptotes omitted but still implied graph crosses or curls away from asymptotes Pre-Calculus Mathematics: Marking Guide (June 2017) 133

138 Appendix B IRREGULARITIES IN PROVINCIAL TESTS A GUIDE FOR LOCAL MARKING During the marking of provincial tests, irregularities are occasionally encountered in test booklets. The following list provides examples of irregularities for which an Irregular Test Booklet Report should be completed and sent to the department: completely different penmanship in the same test booklet incoherent work with correct answers notes from a teacher indicating how he or she has assisted a student during test administration student offering that he or she received assistance on a question from a teacher student submitting work on unauthorized paper evidence of cheating or plagiarism disturbing or offensive content no responses provided by the student (all NR ) or only incorrect responses ( 0 ) Student comments or responses indicating that the student may be at personal risk of being harmed or of harming others are personal safety issues. This type of student response requires an immediate and appropriate follow-up at the school level. In this case, please ensure the department is made aware that follow-up has taken place by completing an Irregular Test Booklet Report. Except in the case of cheating or plagiarism where the result is a provincial test mark of 0%, it is the responsibility of the division or the school to determine how they will proceed with irregularities. Once an irregularity has been confirmed, the marker prepares an Irregular Test Booklet Report documenting the situation, the people contacted, and the follow-up. The original copy of this report is to be retained by the local jurisdiction and a copy is to be sent to the department along with the test materials. 134 Pre-Calculus Mathematics: Marking Guide (June 2017)

139 Irregular Test Booklet Report Test: Date marked: Booklet No.: Problem(s) noted: Question(s) affected: Action taken or rationale for assigning marks: Pre-Calculus Mathematics: Marking Guide (June 2017) 135

140 Follow-up: Decision: Marker s Signature: Principal s Signature: For Department Use Only After Marking Complete Consultant: Date: 136 Pre-Calculus Mathematics: Marking Guide (June 2017)

141 Appendix C Table of Questions by Unit and Learning Outcome Unit A: Transformations of Functions Question Learning Outcome Mark 6 R4, R R R R R5 1 29a) R1 3 29b) R1 1 41a) R1 1 41b) R1 1 49a) R1 1 49b) R1 1 Unit B: Trigonometric Functions Question Learning Outcome Mark 1 T1 2 7 T2, T T T T T T T T3 2 Unit C: Binomial Theorem Question Learning Outcome Mark 2 P3 2 4 P4 3 8 P P P P4 1 Unit D: Polynomial Functions Question Learning Outcome Mark 10 R R R R R12 1 Pre-Calculus Mathematics: Marking Guide (June 2017) 137

142 Unit E: Trigonometric Equations and Identities Question Learning Outcome Mark 5 T T T T5, T T T T6 1 Unit F: Exponents and Logarithms Question Learning Outcome Mark 3 R R R R R R7, R R R7, R8 2 Unit G: Radicals and Rationals Question Learning Outcome Mark 9 R R R R R R R Pre-Calculus Mathematics: Marking Guide (June 2017)